William is 2 times as old as Ben. 35 years ago, William was 9 times as old as Ben. How old is Ben now?
Solution: We can use the given information to write down two equations that describe the ages of William and Ben. Let William's current age be $w$ and Ben's current age be $b$ The information in the first sentence can be expressed in the following equation: $w = 2b$ 35 years ago, William was $w - 35$ years old, and Ben was $b - 35$ years old. The information in the second sentence can be expressed in the following equation: $w - 35 = 9(b - 35)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $b$ , it might be easiest to use our first equation for $w$ and substitute it into our second equation. Our first equation is: $w = 2b$ . Substituting this into our second equation, we get: $2b$ $-$ $35 = 9(b - 35)$ which combines the information about $b$ from both of our original equations. Simplifying the right side of this equation, we get: $2 b - 35 = 9 b - 315$ Solving for $b$ , we get: $7 b = 280.$ $b = 40$.